Bring a large pot of water to a boil.
In the meantime, place the pine nuts in a 12-inch skillet over medium heat. Cook, stirring occasionally, until golden and fragrant, about 2-3 minutes. Transfer to a bowl to cool.
In the same skillet, add the olive oil, shallot, sliced garlic (reserve the minced garlic) and anchovies. Cook over medium heat, stirring occasionally, until the shallots and garlic are fragrant, about 2 minutes. Add the broccoli and season with salt and pepper. Cook, stirring occasionally, until the broccoli is lightly caramelized and tender, about 6 minutes. Add the sherry vinegar and toss to coat. Remove the pan from the heat and keep warm.
Pile the parsley, lemon zest, toasted pine nuts, and the reserved minced garlic clove on a cutting board. Season with salt and pepper. Chop everything together to make the gremolata—the parsley should be fairly well chopped.
Once the water reaches a boil, season it with salt and add the pasta. Cook the pasta according to the package directions until just shy of al dente. Reserve about 1 cup of the cooking liquid, then drain.
Transfer the pasta to the skillet with the broccoli, along with ½ cup of the reserved cooking water. Place the pan over medium heat and cook, stirring, until the water is reduced and coats the pasta, about 1-2 minutes (add more of the water as needed if the dish looks dry). Remove the pan from the heat and add the gremolata, grated Pecorino, and 1-2 big glugs of olive oil. Season with salt and pepper and toss. Serve the pasta with additional Pecorino on the side. If you’d like, sprinkle with a few red pepper flakes.